# Matrix Diophantine equations over quadratic rings and their solutions

## Keywords:

quadratic ring, matrix, $(z,k)$-equivalence of matrices, matrix Diophantine equation, solution of matrix equation### Abstract

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

*Carpathian Math. Publ.*

**2020**,

*12*, 368-375.